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Physica A Statistical Mechanics and its Applications

Exact solutions for the ferromagnetic and antiferromagnetic two-ring Ising chains of spin- 12

Hossein Fakhri, S. Bahareh Seyedein Ardebili

Published Feb 11, 2019• 4 citations
Publisher
DOI
10.1016/j.physa.2019.02.004

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Spin Correlation and Entropy

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A cumulant-like expansion for the entropy of an $N$-spin system is presented. The successive terms in the expansion relate to successively higher orders of statistical association among spins. It is proved that for any Ising system of general dimensionality with ferromagnetic interactions of arbitrary range, the first two terms in the entropy expansion provide a lower bound for the exact entropy. A corollary of the theorem is that the lower-bound property is also valid for any two-sublattice Ising system with antiferromagnetic interactions between sublattices. An example is given which illustrates the fact that the vanishing of the two-spin cumulant (correlation) does not necessarily imply that the spins are statistically independent. The sum of the first two terms in the expansion is compared numerically with the exact entropy of an $N$-spin chain (and also a ring) with nearest-neighbor ferromagnetic or antiferromagnetic Ising interactions. The comparison, which measures the validity of a Kirkwood-type truncation in this context, is favorable only at sufficiently high temperatures.

Physical review. B, Solid state•Apr 1, 1970•7 citations
Abstract An even-site spins- linear chain of cyclic in the nearest-neighbour and next-nearest-neighbour interactions has necessarily a three-ring structure. In this study, the interactions are considered to be short-range through the coupling constants J and K , respectively. For an arbitrary even number of spins- , the exact classical solution of the partition function and the pair correlations between the nearest- and next-nearest-neighbours are calculated in terms of the constants J and K , without using the energy levels and their degeneracy factors. Then, this partition function is used to verify the validity of an ansatz on the allowed energies of the chain and their corresponding degeneracy factors. The influence of next-nearest-neighbour interactions K on the specific heat capacity per spin at the low-temperatures is analysed for the ferromagnetic and antiferromagnetic chains. We especially show that specific heat capacity exhibits the spin frustration when the ferromagnetic and antiferromagnetic orders are demanded via the coupling constants J and K , respectively. Also, the maximal frustration is obtained analytically by two distinct ground levels for even number of spins- .
Energy levels and their degeneracies for two-ring Ising chains of spins-12\frac{1}{2}21​ with NN and NNN couplings: spin frustration of ferromagnetic and antiferromagnetic orders
A cumulant-like expansion for the entropy of an NNN-spin system is presented. The successive terms in the expansion relate to successively higher orders of statistical association among spins. It is proved that for any Ising system of general dimensionality with ferromagnetic interactions of arbitrary range, the first two terms in the entropy expansion provide a lower bound for the exact entropy. A corollary of the theorem is that the lower-bound property is also valid for any two-sublattice Ising system with antiferromagnetic interactions between sublattices. An example is given which illustrates the fact that the vanishing of the two-spin cumulant (correlation) does not necessarily imply that the spins are statistically independent. The sum of the first two terms in the expansion is compared numerically with the exact entropy of an NNN-spin chain (and also a ring) with nearest-neighbor ferromagnetic or antiferromagnetic Ising interactions. The comparison, which measures the validity of a Kirkwood-type truncation in this context, is favorable only at sufficiently high temperatures.